Key Concepts and Formulas

• Complex Number Representation: A complex number can be represented as $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part, and $i = \sqrt{-1}$.
• Modulus of a Complex Number: The modulus (or magnitude) of a complex number $z = x + iy$ is given by $|z| = \sqrt{x^2 + y^2}$.
• Conjugate of a Complex Number: The conjugate of a complex number $z = x + iy$ is given by $\overline{z} = x - iy$.

Step-by-Step Solution

Step 1: Simplify the expression for $z$.

We are given $z = \frac{(1+i)^2}{a-i}$. First, we simplify the numerator: $(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$. So, $z = \frac{2i}{a-i}$.

To express $z$ in the form $x + iy$, we multiply the numerator and denominator by the conjugate of the denominator, which is $a+i$: $z = \frac{2i}{a-i} \cdot \frac{a+i}{a+i} = \frac{2i(a+i)}{(a-i)(a+i)} = \frac{2ai + 2i^2}{a^2 - i^2} = \frac{2ai - 2}{a^2 + 1} = \frac{-2 + 2ai}{a^2 + 1}$.

Separating the real and imaginary parts, we get: $z = \frac{-2}{a^2 + 1} + \frac{2a}{a^2 + 1}i$.

Step 2: Use the given magnitude of $z$ to find $a$.

We are given that $|z| = \sqrt{\frac{2}{5}}$. Using the formula for the modulus, we have: $|z| = \sqrt{\left(\frac{-2}{a^2 + 1}\right)^2 + \left(\frac{2a}{a^2 + 1}\right)^2} = \sqrt{\frac{4}{(a^2 + 1)^2} + \frac{4a^2}{(a^2 + 1)^2}} = \sqrt{\frac{4 + 4a^2}{(a^2 + 1)^2}} = \sqrt{\frac{4(1 + a^2)}{(a^2 + 1)^2}} = \sqrt{\frac{4}{a^2 + 1}} = \frac{2}{\sqrt{a^2 + 1}}$.

Now we set this equal to the given magnitude: $\frac{2}{\sqrt{a^2 + 1}} = \sqrt{\frac{2}{5}}$. Squaring both sides, we get: $\frac{4}{a^2 + 1} = \frac{2}{5}$. Cross-multiplying, we have: $20 = 2(a^2 + 1)$. Dividing by 2, we get: $10 = a^2 + 1$. So, $a^2 = 9$, which means $a = \pm 3$. Since $a > 0$, we have $a = 3$.

Step 3: Find the conjugate $\overline{z}$.

Substitute $a = 3$ into the expression for $z$: $z = \frac{-2}{3^2 + 1} + \frac{2(3)}{3^2 + 1}i = \frac{-2}{10} + \frac{6}{10}i = -\frac{1}{5} + \frac{3}{5}i$.

Now, find the conjugate of $z$: $\overline{z} = -\frac{1}{5} - \frac{3}{5}i$.

Common Mistakes & Tips

• Carefully handle the powers of $i$. Remember that $i^2 = -1$.
• When rationalizing the denominator, make sure to multiply both the numerator and the denominator by the conjugate of the denominator.
• Pay attention to any given conditions, such as $a > 0$, to choose the correct value.

Summary

We simplified the complex number $z$, used its magnitude to find the value of $a$, and then calculated the conjugate $\overline{z}$. The final answer is $\overline{z} = -\frac{1}{5} - \frac{3}{5}i$.

The final answer is \boxed{-\frac{1}{5} - \frac{3}{5}i}, which corresponds to option (B).